This article reviews some integrators particularly suitable for the numerical resolution of differential equations on a large time interval. Symplectic integrators are presented. Their stability on exponentially large time is shown through numerical examples. Next, Dirac integrators for constrained systems are exposed. An application on chaotic dynamics is presented. Lastly, for systems having no exploitable geometric structure, the Borel-Laplace integrator is presented. Numerical experiments on Hamiltonian and non-Hamiltonian systems are carried out, as well as on a partial differential equation.

We compare the performance of two algorithms of computing the Borel sum of a time power series. The first one uses Padé approximants in Borel space, followed by a Laplace transform. The second is based on factorial series. These algorithms are incorporated in a numerical scheme for time integration of differential equations.

Abstract.A time integration method for the resolution of ordinary and partial differential equations is proposed. The method consists in computing a formal solution as a (eventually divergent) time series. Next, the Borel resummation method is applied to deduce an (sectorial) analytical solution. The speed and spectral properties of the scheme are analyzed through some examples. Applications to fluid mechanics are presented.Résumé. On propose une méthode numérique d'intégration temporelle d'équations différentielles ou aux dérivées partielles. Cette méthode consiste d'abordà calculer une solution sous forme de série formelle, dont le rayon de convergence peutêtre nul. Ensuite, la méthode de resommation de BorelLaplace est utilisée pour déduire une solution analytique (dans un secteur) de l'équation. La rapidité et les propriétés géométriques du schéma sont analyséesà travers quelques exemples. Des applications en mécanique des fluides sont présentées.

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